# Set Theory

Topics: Set, Set theory, Empty set Pages: 6 (578 words) Published: October 20, 2014
﻿Venn Diagram
A Venn diagram usually is a drawing, in which circular areas represent groups of items sharing common properties.  The drawing consists of two or more circles, each representing a specific group or set.  This process of visualizing logical relationships was devised by John Venn (1834-1923).

Each Venn diagram begins with a rectangle representing the universal set.  Then each set of values in the problem is represented by a circle.  Any values that belong to more than one set will be placed in the sections where the circles overlap. The universal set is often the "type" of values that are solutions to the problem.  For example, the universal set could be the set of all integers from -10 to +10, set A the set of positive integers in that universe, set B the set of integers divisible by 5 in that universe, and set C the set of elements -1, - 5, and 6.

Universal Set
The Venn diagram at the left shows two sets A and B that overlap.  The universal set is U.

Values that belong to both set A and set B are located in the center region labeled where the circles overlap.   Intersection
This region is called the "intersection" of the two sets.(Intersection, is only where the two sets intersect, or overlap.)

Union
The notation  represents the entire region covered by both sets A and B (and the section where they overlap).  This region is called the "union" of the two sets.(Union, like marriage, brings all of both sets together.)
If we cut out sets A and B from the picture above, the remaining region in U, the universal set, is labeled ,  and is called the complement of the union of sets A and B.

A complement of a set is all of the elements (in the universe) that are NOT in the set.

NOTE*:  The complement of a set can be represented with several differing notations. The complement of set A can be written as

Disjoint sets

two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets.

Null set or Empty set

The Null set or Empty set There are some sets that do not contain any element at all. For example, the set of months with 32 days. We call a set with no elements the null or empty set. It is represented by the symbol { } or Ø . Some other example of null sets are: The set of dogs with six legs. The set of squares with 5 sides. The set of cars with 20 doors. The set of integers which are both even and odd.

Subset
a subset is a set containing some or all members of another set. For example, if the set S is defined as { a, b, c }, then { a }, { a, c } and { a, b, c } are all subsets of S. The symbol ⊆ is used to indicate a subset. So, if S is a subset of T, then S ⊆ T. The symbol is sometimes read as “subset or equal to”, but in general, sets that are equal are subsets of each other. The formal definition for a subset is:

S ⊆ T ↔ ∀x(x∈S → x∈T)
That is, S is a subset of T if and only if every element of S is also an element T.